Metamath Proof Explorer

Theorem ttcss2

Description: The subclass relationship is inherited by transitive closures. (Contributed by Matthew House, 6-Apr-2026)

Ref Expression
Assertion ttcss2 
|- ( A C_ B -> TC+ A C_ TC+ B )

Proof

Step Hyp Ref Expression
1 ttcid 
 |-  B C_ TC+ B
2 sstr 
 |-  ( ( A C_ B /\ B C_ TC+ B ) -> A C_ TC+ B )
3 1 2 mpan2 
 |-  ( A C_ B -> A C_ TC+ B )
4 ttcss 
 |-  ( A C_ TC+ B -> TC+ A C_ TC+ B )
5 3 4 syl 
 |-  ( A C_ B -> TC+ A C_ TC+ B )